Gamma cumulative distribution function

`gamcdf(x,a,b)`

[p,plo,pup] = gamcdf(x,a,b,pcov,alpha)

[p,plo,pup] = gamcdf(___,'upper')

`gamcdf(x,a,b)`

returns the gamma cdf at each of the
values in `x`

using the corresponding shape parameters in
`a`

and scale parameters in `b`

.
`x`

, `a`

, and `b`

can be vectors,
matrices, or multidimensional arrays that all have the same size. A scalar input is
expanded to a constant array with the same dimensions as the other inputs. The
parameters in `a`

and `b`

must be positive, and the
values in `x`

must lie on the interval
`[0 Inf]`

.

`[p,plo,pup] = gamcdf(x,a,b,pcov,alpha)`

produces
confidence bounds for `p`

when the input parameters `a`

and `b`

are
estimates. `pcov`

is a 2-by-2 matrix containing the
covariance matrix of the estimated parameters. `alpha`

has
a default value of 0.05, and specifies `100(1-alpha)`

%
confidence bounds. `plo`

and `pup`

are
arrays of the same size as `p`

containing the lower
and upper confidence bounds.

`[p,plo,pup] = gamcdf(___,'upper')`

returns
the complement of the gamma cdf at each value in `x`

,
using an algorithm that more accurately computes the extreme upper
tail probabilities. You can use the `'upper'`

argument
with any of the previous syntaxes.

The gamma cdf is

$$p=F(x|a,b)=\frac{1}{{b}^{a}\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{a-1}{e}^{\frac{-t}{b}}dt}$$

The result, *p*, is the probability that a
single observation from a gamma distribution with parameters *a* and *b* will
fall in the interval [0 *x*].

`gammainc`

is the gamma distribution with *b* fixed
at 1.